The sum of a man's age and his son is 66. What are their ages if the digits are reveresed ? The teacher said there are 3 answers.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Let the man's age be ab and the son's age be ba.

Now we are given that the sum of their ages is 66.

So ab + ba = 66

=> 10a + b + 10b + a = 66

=> 11a + 11b = 66

=> a + b = 6

If a = 6 , b = 0

If a = 5, b = 1

If a = 4, b = 2

For values of a = 3 , 2 , 1 and 0 the son's age is no longer less than the man.

Therefore the age of the man and the son can be

(60, 6), (51 ,15) and (42 , 24)

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Let the age of the father is presented by the 2 digit number xy.

Then, the son's age is the 2 digit number yx.

==> The fathers age = 10x + y

==. The son's age = 10y + x

But we know that the sum of their ages is 66.

==> 10x + y + 10y + x = 66

==> 11x + 11y = 66

We will divide by 11

==> x + y = 6

Then the sum of the digit must be 6.

Let us determine the positive integers whose sums are 6.

==> 0 + 6 = 6

Then the fathers age = 60

and the son's age = 06

Also, : 2 + 4 = 6

==> The father's age = 42

The son's age = 24

Also, 5 + 1 = 6

==> The fathers age = 51

The son's age = 15

Also, 3 + 3 = 6

==> The father's age = 33

The son's age = 33

But this solution is impossible because the father's and son's age can not be the same.

Then, possible answers are:

( 51 and 15 ) (24 and 42) and ( 60 and 6)

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

Let 10x+y be the fathers age.

The sun's age is in reverse digits by data. So the sun's age = 10y+x.

Naturally father's age  is more than sun's age.

=> 10x+y > 10y +x.

=> 10x+y-10y-x > 0

=> 9x-9y > 0.

=> 9(x-y) > 0.

=> x-y > 0.

 x > y.....(1)

Also the sum of their ages = (10x+y)+(10y+x) = 11(x+y) which is 66 by data.

So 11(x+y) = 66.

We divide both sides of 11(x+y) = 66 by 11:

x+y = 6.....(2)

So x> y and x+y = 6. So we have the choice: x= 3, x= 4, x = 5, x = 6.

Then  the corresponding y values are  y= 3,  y = 2, y = 1 and y = 0.

So  father's age = 33 ,  42,  51, 60.

Sun's age : 33 , 24, 15, 06.

We exclude 33 as it is not practical.

So father and sun's age are one of the pairs:  (42 , 24) , (51, 15) , (60, 06).

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